-
Article
Application of stability region centroidsin robust PI
stabilization of a class ofsecond-order systems
Mohammad Amin Rahimian and Mohammad Saleh Tavazoei
AbstractIn this paper we offer a tuning method for the design of
stabilizing PI controllers that utilizes the stability region
centroid in the controller parameter
space. To this end, analytical formulas are derived to describe
the stability boundaries of a class of relative-degree-one linear
time invariant second-
order systems, the stability region of which has a closed convex
shape. The so-called centroid stable point is then calculated
analytically and the resultant
set of algebraic formulas are utilized to tune the controller
parameters. The freedom to choose the surface density function in
the calculation of
centroid stable point provides the designer with the possibility
to incorporate optimal or robustness requirements in the controller
design process.
The proposed method uses the stability regions in the controller
parameter space to ensure closed-loop stability, and, while
offering robust stability
properties, it does not rely on predetermined information with
regard to the nature or range of parameter variations and
coefficient uncertainty
bounds. Being situated away from the boundaries of the stability
region in the controller parameter space, controllers designed
based on the centroid
method are both robust and non-fragile.
KeywordsNumerator dynamics, optimal tuning, PI controller,
robust tuning, second-order system, stability region, stabilizing
controllers, surface density
Introduction
Favoured for their simple structure, eectiveness and robust-
ness, PID controllers remain of paramount importance in
thecontrol of industrial processes (Astrom and Hagglund,
1995,2005). Moreover, good man-machine interfaces for the
speci-cation of controller structure and parameters, as well as
ecient tuning tools, are necessary requirements for wide-spread
adoption of any industrial controller; and methodsthat can
expedite, ameliorate and simplify the tuning process
are highly sought after (Astrom and Hagglund, 2001).Among
various design criteria to be satised by a control-
ler, closed-loop stability is the most basic, and the stability
of
nal closed-loop systems for both fractional-order and
inte-ger-order PID controllers have been the subject of many
pre-vious studies (Nusret Tan et al., 2006; Hamamci, 2007; Fang
et al., 2009). The set of controller parameters yielding a
stableclosed-loop system is known as the stability region, and a
lotcan be learned about PID control through analysis of
stabilityregions. The problem of nding all stabilizing PID
controllers
for a given plant, which leads to the stability domains in
thecontrollers parameter space, has traditionally been dealt
withusing Nyquist plot, stability boundary locus,
characteristic
equation and frequency-based methods. Applications of
theHermiteBiehler theorem (Caponetto et al., 2010), as well
aselaborate polynomial calculations (Hohenbichler and
Ackermann, 2003a; Soylemez and Baki, 2003;Hohenbichler, 2009),
have been the driving force behindsome of the recent results on
this topic. The author in
Hamamci (2008) has introduced a method for investigating
the stabilization of fractional-order PI controllers that is
basedon plotting the global stability region in the (kp, ki) plane.
A
similar method is adopted by the authors in Hamamci andKoksal
(2010) to derive the stability regions for fractional-order PD
controllers in the (kp, kd) plane. The authors inRahimian and
Tavazoei (2010a,b) have provided an extension
of the method used in Hamamci (2008) to plot the
stabilityregions for the integer-order approximations of PIl and
PDm
controllers. The same methods form the basis of the design
scheme used in Rahimian et al. (2010) for the stabilization
oftwo-mass drive systems with elastic coupling.
The main theme of this paper is the calculation of the cen-
troid point for the stable region in the controller
parameterspace. This so-called centroid stable point serves as the
designchoice, according to which controller parameters are set.
The
centroid stable point has the advantage that it lies at the
farth-est possible distance from the boundaries of the stability
regionand will therefore ensure the robust stability of the
resultingclosed-loop control system. Accordingly, while oering
robust
stability properties, the proposed method does not rely on
anyinformation pertaining to the nature or range of
parametervariations. Furthermore, the freedom to choose the
surface
Electrical Engineering Department, Sharif University of
Technology, Iran
Corresponding author:
Mohammad Saleh Tavazoei, Electrical Engineering Department,
Sharif
University of Technology, Tehran, Iran
Email: [email protected]
Transactions of the Institute of
Measurement and Control
34(4) 487498
The Author(s) 2011Reprints and permissions:
sagepub.co.uk/journalsPermissions.nav
DOI: 10.1177/0142331211400117
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density in the calculation of centroid stable point provides
amore exible tuning strategy, and therefore an easier way toachieve
control requirements such as optimality or robustness.
The remainder of this paper is organized as follows. Themethod
of Hohenbichler and Ackermann (2003b), Hamamci(2008) and Hamamci
and Koksal (2010) is used in the nextsection to derive an
analytical description for the boundaries
of the stability region in a closed-loop system comprised of aPI
controller and a strictly proper second-order plant
transferfunction. Next, in the section titled Centroid stable
point
and the proposed tuning formulas, a set of conditions areset
forth under which the stability region described in its pre-ceding
section has a closed convex shape, for which the cen-
troid exists and is meaningful stability-wise. Following
thespecication of constraints on the plant parameters, a con-stant
value for the surface density function is assumed and the
centroid of the stability region is analytically calculated.
Thesection titled Choice of surface density for robust and
optimaltuning elaborates on the eects of the choice of surface
densityon the calculation of the centroid stable point, and
illustrates
how various surface density functions can be harnessed to
pro-vide the designed control system with robustness and
optimalityproperties. To this end, a specic example, where surface
den-
sity function is adjusted to procure optimal disturbance
rejec-tion properties, is discussed analytically, and algebraic
formulasfor the corresponding centroid stable point are proered.
Using
constant and tailored surface density functions, two choices
ofcentroid stable points are computed for an example system
insections Centroid stable point and the proposed tuning for-mulas
and Choice of surface density for robust and optimal
tuning respectively, and the corresponding closed-loop sys-tems
are simulated and compared in the section titledSimulation results
and discussion. A second example in the
latter section compares the performance of the proposed
tuningrules with classical ZieglerNichols rules. Lastly, some
conclud-ing remarks are provided in the nal section concerning
the
scope and possible extensions of the proposed approach.
Stability regions for the PI controller
Consider the basic closed-loop control system depicted inFigure
1, where y is the output and r is the reference input.
The inputoutput relation for the closed-loop system of
Figure 1 in the Laplace domain is given by
YsRs
CsGs1 CsGs , 1
with G(s) and C(s) indicating the plant and compensator
transfer functions, respectively; the characteristic
equation
for the closed-loop system will be derived from setting
thedenominator of (1) to zero, and is as follows:
1 CsGs 0: 2
The numerator of (2) is called the characteristic polynomialand
is denoted by P(s). This paper focuses on the problem of
controlling a general second-order plant given by
Gs as bs2 cs d , 3
with a PI controller given by
Cs kp kis: 4
Numerous real-world processes can be shown to observe
thetransfer function in (3), and second-order processes
withnumerator dynamics are the focus of Seborg et al. (2004:
Section 6.1). There, it is explained that second-order
systemswith right-half-plane (RHP) zeros exhibit the
inverse-response phenomenon, where the direction of an initial
response to a step input contradicts that of its nal
steadystate. The authors in Seborg et al. (2004) then describe
twopractical scenarios, involving the liquid levels in a
distillation
column and the temperature of a chemical reactor withexothermic
reactions. In both cases, competing dynamiceects that operate on
two dierent time scales lead to theinverse-response behaviour.
Accordingly, second-order sys-
tems with numerator dynamics that exhibit inverse-responseor
overshoot behaviour can occur whenever two physicaleects act on the
process output variable in dierent ways
and with dierent time scales.Another classical example of a
second-order process with
numerator dynamics is a mass-spring-dashpot (MSD) system
in series, where a mass, m, is attached to a spring with
con-stant k, which is attached to a dashpot with damping coe-cient
h, which is attached to a wall. Accordingly, the transfer
function from the force F, acting on the mass m, to its speed
n,is given by
nsFs
hs kmhs2 mks kh : 5
Moreover, several of the plants that are investigated in the
literature are formulated in form of (3). These include
thetransfer function from motor torque to load torque in atwo-mass
drive system with elastic coupling in Rahimian
et al. (2010), and the transfer function from steeringangle to
tilt angle for the unrideable bicycle in Klein(1989). Other
examples from robotics include the free loadlinear actuator model
for the biomimetic robot in Robinson
et al. (1999) and the transfer function of the end
eectormotion/command position for the single-link manipulatormodel
in Xu and Paul (1988). The transfer function
from the elevator deection (de) to the pitch rate (q) inight
control systems (Oosterom and Babuska, 2006;El-Mahallawy et al.,
2011) also follows the same dynamics
as described by (3).
r C(s) G(s) y+
Figure 1 A basic closed-loop control system, using unity
negative
feedback.
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In addition to the aforementioned examples, in which realworld
plants are modelled by the same type of transfer func-tions as (3),
there are several cases where approximation,
order-reduction or identication methods lead to a formula-tion
of the process dynamics that is in accordance with (3).The transfer
function for every First-Order Plus Dead-Time(FOPDT) system, in
which the time delay is replaced by a
rst-order Pade approximation, takes the form of a second-order
system with a non-minimum phase zero in the numera-tor, as does the
transfer function for every Second-Order Plus
Dead-Time (SOPDT) system, in which the exponential term
isapproximated by the rst two terms in its Taylor series. Theauthor
in Luus (1999) introduces a method to optimally
choose the coecients for a second-order reduced model,the
transfer function for which is given by (3). The optimiza-tions are
performed in the frequency domain, using a multi-
pass optimization technique, and the resulting
second-orderreduced models can closely follow the Nyquist plots of
theoriginal fth- and eighth-order systems. Last, but not
least,plant models of the type in (3) can be the result of an
identi-
cation procedure. The authors in Wang and Chen (2009)
usesubspace system identication methods to determine thetransfer
function matrices for the Multi-Input Multi-Output
dynamics of a proton exchange membrane fuel cell (PEMFC)system.
The resulting transfer functions, after being convertedinto
continuous-time by zero-order-hold, are all in the
form of (3).According to (2), the characteristic polynomial can
be cal-
culated as
Ps s3 c akps2 d aki bkps bki: 6
The Hurwitz stability criterion for the characteristic
polyno-
mial in (6) is satised if and only if every root of P(s) lies
inthe left half plane (LHP). In other words, the imaginary axisand
the origin are the only places where the stability shift of
the system will occur (Fang et al., 2009). However, the
posi-tions of roots of P(s) will change continuously as long as
itscoecients are continuous functions of the plant or
controller
parameters and those parameters are changed continuously.Thus, a
stable polynomial, P(s), whose roots all lie in theLHP, becomes
unstable if and only if at least one root crossesthe imaginary
axis. Accordingly, the set of coecients that
would lead to a stable closed-loop system can be denoted
asstability domains in the parameter space of the
characteristicpolynomial P(s). These stability domains are
determined by
the following three boundaries, which describe the rootscrossing
from the LHP to the RHP and vice versa(Hohenbichler and Ackermann,
2003b; Hamamci, 2008;
Hamamci and Koksal, 2010):
Denition 1. The Three Root Boundaries:
(a) Real Root Boundary (RRB): A real root crosses over
theimaginary axis at the origin (s 0), and for P(s) given by(6) it
can be determined as
bki 0! ki 0, 7
i.e. by setting the constant term to zero.
(b) Complex Root Boundary (CRB): A pair of complex rootswill
cross over the imaginary axis at s jv, and their cor-responding
locations in the parameter space are obtained by
substituting s jv in P(s) and setting its real and
imaginaryparts to zero.
(c) Innite Root Boundary (IRB): The last possibility is for
aroot to cross the imaginary axis at innity (|s|!), and itcan be
determined by setting the coecient for the termwith the largest
power of s in (6) to zero. Here, since theterm with the largest
power corresponds to s3 and its coe-
cient cannot be zero, the IRB does not exist.
Once the coecients of (6) are described in terms of con-
troller parameters, the three boundaries given by Denition 1will
translate into stability boundaries in the parameter spaceof the PI
controller, i.e. {kp, ki}. These stability boundaries
separate dierent regions in the parameter space, and so,
todetermine the stability of a given region, it suces to checkthe
stability of one test point within that region (e.g. by theNyquist
criterion or through investigation of the roots of the
corresponding characteristic polynomial).Next, substituting s jv
in (6) and equating the real and
imaginary parts to zero results in the following two
equations,
respectively:
bki c akpv2, 8
v2 d bkp aki: 9
Eliminating the parameter v between the two equations in (8)
and (9) yields
ki cd cb ad kp abkp2
b ca a2kp : 10
In light of the previous discussion, the stability boundaries
for
the control system described by Figure 1 and equations (3)and
(4) are given by equations (7) and (10), which specify theRRB and
CRB, respectively.
Lastly, in order to determine the stability regions in
thecontroller parameter space, we should test the stability
ofsingle test points in each of the regions generated from
theintersection of the boundaries given by (7) and (10).
Example 1. An unstable non-minimum-phase
second-orderrelative-degree-one linear time invariant (LTI)
system.
In order to demonstrate the aforementioned procedure,the
following unstable non-minimum-phase
second-orderrelative-degree-one LTI system, which is taken from
Roup
and Bernstein (2003: Example 1), is considered:
G1s s 5s2 15s 5 : 11
The sample plant in (11) can be derived from (3) by setting
a 1, b5, c 15 and d5, and it is used throughout therest of the
paper to demonstrate the applicability of the pro-posed method.
Having an unstable pole at p 0.3262 and anon-minimum phase zero at
z 5, the system in (11) is a
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typical hard-to-control second-order plant (Astrom, 2000).The
presence of the RHP zero imposes a fundamental limita-tion on
control, and high controller gains will induce closed-
loop instability (Skogestad and Postlethwaite, 1996). The
sta-bility region for (11) can be computed using the boundaries
in(7) and (10), and is depicted in Figure 2. The closed andconvex
shape of the stability region in Figure 2, which occu-
pies a limited portion of the plane, is a further indication
thatthe underlying closed-loop system is not well-behaved and
ishard to control.
In the next section we nd the centroid of the stabilityregion
depicted in Figure 2 analytically, and derive a set ofalgebraic
formulas that can be used as a suitable choice for
the unknown controller parameters, kp and ki. Such a choicehas
the advantage that it lies at the farthest possible distancefrom
the boundaries of the stability region and would there-
fore ensure the robust stability of the resulting
closed-loopcontrol system.
Centroid stable point and the proposedtuning formulas
This section focuses on the calculation of stability region
cen-troids. This is motivated by the fact that the geometric
centreof an objects shape can be a best option if the aim is to
avoid
its boundaries and exterior as much as possible. This is
trueprovided that the underlying object has a closed convexshape.
Accordingly, prior to the calculation of centroidstable points, a
set of constraints on the plant parameters a,
b, c and d are needed to ensure that the stability regions
haveindeed a closed convex shape, as was the case with Figure 2,for
which the stability region was an upward semi-parabola,
capped at the top by the kp axis. These conditions ensure
that
calculation of the centroid for the stable region is justiedfrom
the perspective of closed-loop system stability. Suchsystems, for
which the stability region is bounded, would in
eect be hard to control, and examples often include
planttransfer functions that are unstable or non-minimum phase.
Peering into (10), it will dawn on us that the CRB crossesthe kp
axis at the following two points:
kpr1 c
a, 12
kpr2 d
b, 13
and it has a vertical asymptote at
kpa b ac
a2: 14
Depending on the relative locations of the two roots in
(12) and (13) and the asymptote in (14), several cases mayarise,
each of which is realized under a particular set of con-ditions on
the plant parameters. Among the many possibili-
ties, those specied in Tables 1 to 4 culminate in
stabilityregions with a closed convex shape. The conditions inTable
1 and Table 2 yield a downward semi-parabola that
is capped at the bottom by the kp axis, while those inTable 3
and Table 4 yield an upward semi-parabola that iscapped at the top
by the kp axis. Table 1 and Table 3 corre-spond to the cases where
ab< 0, while Table 2 and Table 4 listthe conditions for the
cases where ab> 0. The plant numera-tor dynamics in the former
case include a non-minimumphase zero, leading to closed-loop
systems that are particu-
larly hard to control. Each table lists the additional
40 30 20 10 0 10 20400
350
300
250
200
150
100
50
0
50
100
kp
k i
CRBRRB
Stability region
Figure 2 The stability region is plotted for the control system
of Figure 1, where G(s) and C(s) are given by (11) and (4),
respectively.
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constraints that should be satised for various combinationsof
signs of the parameters c and d.
The analytical formulas derived for the CRB in the
previoussection provide us with the possibility of selecting a
point inside
the stable region that is at the farthest possible distance
fromevery point on the boundary. The coordinates (xc, yc) for
thecentroid of a graph in the (x, y) plane are given by
xc R R
sx, yx dxdyM
, 15
yc R R
sx, yy dydxM
, 16
where M is given by
M Z Z
sx, y dydx, 17
and s(x, y) denotes the surface density function (Thomas
andFinney, 1999). The double integrals in (15), (16) and (17)
are
calculated over a stable region such as the one in Figure
2.Under the assumption that no information is available as tothe
range or type of variations against which the closed-loop
system should exhibit robustness, the surface density s(x, y)
isset to be a constant number and is thus eliminated from
thenumerators and denominators of (15) and (16). Using (7) and
(10) for the boundaries of the stability region, and
assumingthat one of the condition sets in Tables 1 to 4 are
satised, thecentroid stable point for the closed-loop system of
Figure 1,with the plant and controller given by (3) and (4)
respectively,
can be calculated analytically. The results are the
followingalgebraic tuning rules:
kpc I1
Mc, 18
kic I2
Mc, 19
where I1, I2 and Mc are given by
I1 S33a4b2
PS22a5b2
RS1a6
ac ba2
Q, 20
I2 S36a5b
PS22a6b
P2 3b2RS1
2a7b R b2
a3Q, 21
Mc S22a3b
S1Pa4b
Q, 22
and P, Q, R and Sn are dened as
P a2d b2, 23
Q bRa5
ln Rb2
, 24
R P abc, 25
Sn andn bncn, n 2 N: 26
The algebraic tuning formulas introduced in (18) and (19)
can
be applied to Example 1 and the corresponding centroidstable
point can be computed as
kpc1 8:8929,
kic1 8:9291: 27
Figure 3 denotes the centroid stable point along with
thestability region for the system in (11).
Here it should be noted that the proposed choice of thestability
region centroid as the tuning rule for setting control-ler
parameters is inherently conservative. The centroid stable
point tuning rules in (18) and (19) are, in fact, a sucient
butnot necessary condition for closed-loop system stability,
andusing them as the basis for design is justied only when sta-
bilization is the rst and foremost criterion that is expected
tobe satised by the control system. Such conservatism in
Table 3 Constraints on the plant parameters for the case where
a> 0
and b< 0
Signs of parameters c and d Additional conditions
c< 0, d> 0 baca2\ db \ ca or ca \ db
c> 0, d< 0 baca2\ db \ ca or ca \ db
c< 0, d< 0 baca2\ db
c> 0, d> 0 None
Table 1 Constraints on the plant parameters for the case where
a< 0
and b> 0
Signs of parameters c and d Additional conditions
c< 0, d> 0 ca \ db \ baca2 or db \ cac> 0, d< 0 ca \
db \ baca2 or db \ cac< 0, d< 0 db \ baca2c> 0, d> 0
None
Table 4 Constraints on the plant parameters for the case where
a> 0
and b> 0
Signs of parameters c and d Additional conditions
c> 0, d> 0 ca \ db \ baca2 or db \ cac< 0, d< 0 ca \
db \ baca2 or db \ cac> 0, d< 0 db \ baca2c< 0, d> 0
None
Table 2 Constraints on the plant parameters for the case where
a< 0
and b< 0
Signs of parameters c and d Additional conditions
c< 0, d< 0 baca2\ db \ ca or db \ ca
c> 0, d> 0 baca2\ db \ ca or db \ ca
c> 0, d< 0 baca2\ db
c< 0, d> 0 None
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design is best justied in the case of highly uncertain
systems,where plant parameters can only be determined with a
limitedprecision and are always subject to variations. Lack of
infor-mation with regard to the nature and range of such
variations
is a second factor which favours the use of the stability
regioncentroid as the tuning point. Another class of systems
forwhich the tuning rules of (18) and (19) are particularly
useful is the class of highly unstable systems with
narrowstability regions, where it becomes critically important
toavoid the boundaries of the stability region. Being situated
away from the boundaries of the stability region in the
con-troller parameter space, controllers designed based on
thecentroid method are both robust and non-fragile.
Accordingly, such controllers can tolerate certain variationsin
the parameters of both the controller and the plant (Keeland
Bhattacharyya, 1997; Makila et al., 1998). The
parameteruncertainties can be due to either inherent physical
properties
or modelling diculties, and having control systems whichremain
operational in the face of parameter variations isdesirable in both
cases.
In the next section, the eects of the choice of surfacedensity
function on the calculation of the centroid stablepoint in (15) to
(17) is investigated. It is further demonstrated
through an example that the choice of surface density func-tion
can, in fact, be leveraged to produce desirable optimalityor
robustness properties.
Choice of surface density for robust andoptimal tuning
In the previous section it was pointed out that in the absenceof
any extra information the surface density function in (15),
(16) and (17) is considered to be a constant number, which
is
then cancelled out in the calculations. The uniform density,in
eect, corresponds to the case where robust stability is tobe
guaranteed and no extra information is available withregard to
variations of plant parameters and coecient
uncertainty bounds. This, however, need not necessarily bethe
case. For instance, if we know how the shape of thestability region
is aected by variations in the plant para-
meters, then we might be able to place the centroid so that itis
properly distanced from sensitive locations. Accordingly,the
undesirable locations are discriminated against by being
assigned lower weights when designating the surface
densityfunction across the stability region. The following
para-graphs elaborate on the choice of surface density function
and how it can be modied to satisfy various
designspecications.
By and large, any tuning formula can be thought of as acentroid
stable point which is derived through (15), (16) and
(17) with an appropriate choice of surface density function s.In
general, the surface density function will be adjusted sothat less
weight is given to the undesirable regions and the
centroid stable point is drawn toward more favourableregions
because of their higher weights. To clarify thispoint, an example
is given in which the surface density func-
tion is designed to provide the tuned control system withoptimal
disturbance rejection characteristics.
Tuning methods based on an optimization approach havereceived
considerable attention in the literature
(Panagopoulos et al., 2002). The problem of designing anoptimal
PI controller, C*(s), given by (4), can be formulatedas determining
the optimal parameters kp
and ki so that aset of constraints are satised and a cost
function J(kp, ki) isminimized. In general, the parameter
optimization designprocess consists of searching the space of
variable controller
parameters as a function of some performance index J to
40 30 20 10 0 10 2030
20
10
0
10
20
30
40
50
60
kp
k iCRBCentroid stable pointRRB
Figure 3 The centroid stable point is depicted for the control
system given by Figure 1 and equations (11) and (4).
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determine where the performance index is minimized
(Ogata,2009).
The occurrence of eective constraints which necessitate
theoptimal solution to reside at the boundaries of the
admissibleparameter domain would mean that there is no guarantee
forthe necessary control requirements to remain satised under
various parameter variations and coecient uncertainties.Using
the centroid approach, however, the centroid point isguaranteed to
maintain a safe distance from admissible
region boundaries, while the optimality properties are moreor
less preserved through the appropriate choice of surfacedensity
function s(kp, ki). Such a choice of s(kp, ki) would
have to be a positive decreasing function of the cost J so
thatthe regions with higher values of J are discriminated
against.The positive s would ensure that the centroid is within
the
admissible region, provided that it has a closed convex shape.In
eect, it suces fors to not change sign across the region forwhich
the centroid is being calculated. This is due to the factthat the
actual positive or negative sign of a uni-sign function s
is cancelled out from the numerator and denominator of both(15)
and (16), and hence it does not aect the nal result.
The freedom in choosing the function s can be a valuableasset in
the management of the important trade-o betweenperformance and
robustness in control systems (Kristiansson
and Lennartson, 2002). On the one hand, aggressive choicesof s,
which decrease drastically with increasing J, would cul-minate in
centroids that are closer to the truly optimal solu-
tions. Such aggressive choices, on the other hand, maycompromise
the closed-loop system robustness by being situ-ated too close to
the admissible region boundaries. In otherwords, with the choice of
s, the designer can decide to move
away from a conservative design and closer to an optimalone, or
vice versa.
Examples of decreasing functions which may be used todescribe
the surface density function in terms of a positivecost function J
include
s 1JN
, N 1, 2, 3, 4, . . . , 28
where lower values of N correspond to more conservativechoices,
and higher values of N are chosen when aggressively
optimal solutions that are near the truly optimal one
arepreferred. Depending on the relative importance of robust-ness
or optimality properties, the power of J in the denomi-
nator of (28) may be decreased or increased. In the
followingparagraphs an optimal tuning scenario is considered, in
whichs is chosen as
s 1J: 29
Once the cost function J in (29) is described in terms of
con-troller parameters kp and ki, the function ss*, given by(29),
can be used in the context of (15) to (17) to calculate
the corresponding centroid stable point. Integrated Error (IE)is
an integral performance index, which is dened (Astromand Hagglund,
1995) as
IE Z 0
et dt, 30
where e(t) is the error to a step input function. If the
closed-loop system output is denoted by y(t), then e(t) is given
by
et 1 yt, 8t>0: 31
Here, the cost function to be minimized is chosen as
follows:
J jIEj: 32
The integral performance index in (30) can be used to evalu-ate
the performance of a designed control system or, as here,
for optimal tuning of xed structure controllers. In the
lattercase, the parameters of the control system are optimized
byminimizing the integral performance index given by (32). In
Astrom and Hagglund (1995) it is shown that for a
stableclosed-loop system, if the error is initially zero and a
unitstep disturbance is applied at the plant input, then
IE 1ki: 33
Accordingly, the closed-loop system disturbance rejection canbe
optimized by minimizing the IE criterion, and it is a nat-ural
choice for the control of quality variables for processes
where the product is sent to a mixing tank (Astrom andHagglund,
1995). Several PID tuning rules based on minimi-zation of the IE
criterion have been presented, and some
incorporate gain and phase margin specications to
ensurestability and robustness of the controlled systems (Astromet
al., 1998). The authors in Hwang and Hsiao (2002), forinstance,
present an eective solution to the non-convex opti-
mization problem, which arises in the design of stabilizing
PIand PID controllers based on the minimization of the integralof
the error caused by a step disturbance and subject to con-
straints on maximum sensitivity Ms.
Using the results in (29), (32) and (33), we can write
skp, ki 1J jkij: 34
Next, supposing that one of the condition sets in Tables 1 to
4
is satised, the surface density function given in (34) can
beused in the context of (15) to (17) to calculate the
centroidstable point for the stability boundaries given by (7) and
(10).The resultant centroid stable point is given by
kp I1
I2, 35
ki I2
I2, 36
where I2 is given by (21). I1 and I2 in (35) and (36) are
given by
I1 f1 kpr2
f1 kpr1 , 37I2 f2 kpr2
f2 kpr1 , 38
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where kpr1 and kp
r2 are given by (12) and (13), and the func-tions f1() and f2()
are dened in the Appendix.
The algebraic tuning formulas introduced in (35) to (38)
can be applied to Example 1, and the corresponding
optimalcentroid stable point can be computed as
kp1 9:2808,
ki1 12:8080: 39
Figure 4 denotes the optimal centroid stable point given by
(39), along with the centroid stable point in (27) computed
forthe system in Example 1.
In the next section the two choices of controllers given by
the two centroid points computed in this and the
previoussections for the sample plant (11) are simulated, and the
per-formance of the corresponding closed-loop systems are com-
pared. A second example compares the performance of twoPI
controllers: one is designed using the proposed centroidscheme, the
other is tuned using the ZieglerNichols fre-quency domain
method.
Simulation results and discussion
The output responses of the closed-loop system of Figure 1with
the sample system in (11) and the two controllers givenby (27) and
(39) are plotted in Figure 5. The results show that,
being situated at a farther distance from the stability
regionboundaries, the controller in (27) yields a less
oscillatoryresponse to the unit step input applied at t 0. The
controllerin (39) is, however, more eective in the rejection of the
unit
step disturbance which is applied to the plant input at t 5.The
Integrated Error (IE) is a common performance index,
which is dened as in (30). The value of IE for a unit
stepdisturbance has been calculated in Astrom and Hagglund(1995)
and is given by (33). Accordingly, the corresponding
values for the controllers in (27) and (39) are 0.1120
and0.0781. This conrms that the integral of the error causedby the
unit disturbance at t 5 has a lower absolute value forthe
controller in (39). This is to be expected, since the con-
troller in (39) has been designed for an optimal performancewith
regard to the IE performance index, and for its distur-bance
rejection.
Next, in order to compare the robust stability of the
twosystems, uncertainties are introduced in plant parameters aand c
and their eects are investigated. First, a is increased
from its nominal value of a 1 until the corresponding systemis
no longer stable. The closed-loop system with the controllerin (27)
destabilizes at a 1.5, whereas the one with the sub-optimal
controller given by (39) becomes unstable at a 1.34.This conrms the
superior robustness of the centroid stablepoint as compared to its
optimal version. Repeating theexperiment by decreasing c from its
nominal value of c 15results in a similar observation. The
closed-loop systems withthe controllers in (27) and (39)
destabilize at c 10.37 andc 11.51 respectively. It is worth
highlighting that the trueoptimal solution, which minimizes the
cost function in (32)subject to the stability condition,
corresponds to the base ofthe downward semi-parabola in Figure 4,
and thus lies on the
boundary of the stability region.Last, but not least, it is
worth mentioning that the presence
of a non-minimum phase zero at z 5 for the plant in (11)imposes
inherent restrictions on the disturbance rejection
properties of the closed-loop system (Astrom, 2000). Due tothe
so-called push-pull eect, low-frequency non-minimum
40 30 20 10 0 10 2030
20
10
0
10
20
30
40
50
60
kp
k i
Centroid stable points
s = constant
s = 1J
= |ki|
Figure 4 The two centroid stable points, derived using a
constant and an optimal s, are depicted above.
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phase zeros inhibit superior disturbance rejection at low
fre-quencies (Freudenberg and Looze, 1998). Accordingly,increasing
the parameter a from its nominal value of a 1causes the zero at z
5a to move closer to the origin, thusfurther degrading the
disturbance rejection at low frequen-cies. The next example
compares the performance of the
proposed tuning rules with classical ZieglerNichols rulesfor an
inverse response process.
Example 2. PI-control of an inverse response process.Here,
Skogestad and Postlethwaite (1996: Example 2.3) is
considered and the performance of the proposed centroid
0 20 40 60 80 100 1200.2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
t (s)
y(t)
Unit step responses
Ziegler and NicholsCentroid stable point
Figure 6 The responses for the closed-loop system of Figure 1,
with (40) as the plant and the two PI controllers specified by (41)
and (42).
0 2 4 6 8 101
0.5
0
0.5
1
1.5
2
2.5
t
y(t)
Unit step and disturbance responses
s = constant
s = 1J = |k i|
Figure 5 The responses for the closed-loop system of Figure 1,
with (11) as the plant and the two PI controllers specified by (27)
and (39).
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stable point controller is compared with a PI controller that
isdesigned according to the classical tuning rules of Ziegler
andNichols. The plant model (time in seconds) is given by
G2s 32s 15s 110s 1 : 40
The ZieglerNichols PI controller parameters, which arederived
from the ultimate gain and ultimate period accordingto the
frequency response method (Astrom and Hagglund,
1995), are set as follows (Skogestad and Postlethwaite,
1996):
kpZN 1:1400,
kiZN 0:0898: 41
The algebraic tuning formulas introduced in (18) and (19) canbe
applied to the plant in (40) and the corresponding centroidstable
point can be computed as
kpc2 1:1563,
kic2 0:0730: 42
The output responses and control signals for the
closed-loopsystem of Figure 1 with the sample system in (40) and
the twocontrollers given by (41) and (42) are plotted in Figures 6
and
7 respectively. It is important for the control signals not
toviolate the actuator and plant dynamics due to the presence
ofnonlinearities such as saturation.
The presence of integral action in form of a PI
controllerremoves the steady-state oset in the step response of
theclosed-loop system. Table 5 compares various performance
measures for the two control systems and their correspondingstep
responses. The results conrm that the ZieglerNicholstuning rules
are somewhat aggressive, culminating in a closed-loop system with
smaller stability margins and a more oscil-
latory response (Skogestad and Postlethwaite, 1996). On theother
hand, the conservative nature of the centroid tuningrules leads to
a closed-loop system with larger stability mar-
gins and a superior transient response. The conservative
rules,proposed in (18) and (19), are particularly useful for
thetuning of systems with narrow stability regions, such as the
one in Example 2, for which it is critically important to
avoidthe boundaries.
Conclusion
This paper oered a set of algebraic tuning formulas thatwould
ensure robust stability of the closed-loop system without
0 20 40 60 80 100 1200.4
0.2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
t (s)
u(t)
Control signals for a unit step reference input
Ziegler and NicholsCentroid stable point
Figure 7 The control signals for the closed-loop system of
Figure 1, with (40) as the plant and the two PI controllers
specified by (41) and (42).
Table 5 Comparison of the performance measures for centroid
stable
point and frequency domain ZieglerNichols tuning rules
Performance measure Centroid ZN
Phase margin 22.93558 19.30198
Gain margin 1.7250 1.6298
Phase crossover frequency 0.353 rad/s 0.3375 rad/s
Gain crossover frequency 0.2363 rad/s 0.2370 rad/s
Rise time (0 90%) 8.0669 s 7.9916 sSettling time (65%) 52.7526 s
65.2532 s
Overshoot 53.5915% 63.0855%
Decay ratio 0.3046 0.3602
Maximum sensitivity (Ms) 3.4579 3.9373
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exploiting any prior knowledge as to the nature or range
ofparameter variations in the plant or controller. The
controllerparameters are chosen so that the designed system will
lie at
the farthest possible distance from every point on the bound-ary
of the stability region in the controller parameter
space.Analytical formulas for the stability regions of a
generalsecond-order plant were derived and the desired
controller
parameters were selected as the centroid of the
stabilityregions. Moreover, a set of conditions were
introducedwhich will ensure that the stability regions have a
closed
convex shape, and the calculation of centroids is
thereforemeaningful stability-wise. Unlike classical robust
stabilizationtechniques, the stability region centroid approach
proposed in
this paper does not require the coecient uncertainty boundsto be
known or satisfy any inequality constraints. The con-servative
nature of the proposed tuning rules can prove par-
ticularly useful in the control of closed-loop systems
withnarrow stability regions or highly uncertain parameters.
Thedevised scheme, which is developed for PI controllers and aclass
of second-order relative-degree-one LTI plants, can be
further extended to PD and PID controllers as well as to
thegeneral case of controllers for which the stability region has
aclosed convex shape. In the case of PID and other three-para-
meter controllers, however, the design parameter spaceinvolves
three unknowns (kp, ki, kd) and the resulting calcula-tions, which
involve the centre of mass for a three-dimen-
sional gure, are analytically cumbersome.
Acknowledgment
The authors would like to thank Samira Rahimian for her help
with
calculation of centroids and derivation of conditions in Tables
1 to 4.
Funding
This research received no specic grant from any funding agency
in
the public, commercial or not-for-prot sectors.
Conflict of interest
None declared.
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Appendix
The centroid stable point with s*(kp, ki) |ki|Functions f1() and
f2(), which appear in the formulation ofthe optimal centroid stable
point in (37) and (38), are denedas follows:
f1x b2x4
8a2 P2x2
4a6 x33a4
bR2
2a10gx ac b
l x
hx bPRa6
kx,
f2x b3x4
12a3 P3x
3a9 bR
3
6a11l2x b2Px3
3a5 bP2x2
2a7
bP2Rgxa11
b3R2a11
gx ac bl x
bahx
b2PR2a11l x
2b2PRa7
kx,
where P and R are dened in (23) and (25), and l(x), g(x),
h(x)and k(x) are given by
l x a2x ca b,gx ln l x ,hx b2R
a10a2c2 b2 2abc gx 0:5x2a4 xa2ac b ,
kx xa2 ac b
a4gx:
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